POP

G3.10 Converting Units & Compound Measures

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01 — Explanation

How do I convert between metric length, mass, and capacity units?

Memorize the core factors, all powers of 10:

	Relationship
Length	$1$ cm $=10$ mm, $1$ m $=100$ cm, $1$ km $=1000$ m
Mass	$1$ kg $=1000$ g, $1$ tonne $=1000$ kg
Capacity	$1$ L $=1000$ ml

Converting to a smaller unit always gives a bigger number (multiply); converting to a bigger unit gives a smaller number (divide). For big jumps, convert in stages: $12{,}300$ cm to km goes cm $\to$ m ( $\div100$ ) then m $\to$ km ( $\div1000$ ).

02 — Explanation

Why can't I use the same factor for area and volume conversions?

Because area is 2-dimensional and volume is 3-dimensional — the linear factor must be squared or cubed.

A $1$ m $\times 1$ m square has area $1$ m², which is $100$ cm $\times 100$ cm $=10{,}000$ cm². So the area factor is $100^2$ , not $100$ . A $1$ m cube has volume $1$ m³ $= 100$ cm $\times100$ cm $\times100$ cm $=1{,}000{,}000$ cm³ — the volume factor is $100^3$ .

1 \text{ cm}^2 = 100 \text{ mm}^2 \qquad 1 \text{ m}^2 = 10{,}000 \text{ cm}^2

1 \text{ cm}^3 = 1000 \text{ mm}^3 \qquad 1 \text{ m}^3 = 1{,}000{,}000 \text{ cm}^3

Common Misconception: Using the linear factor (e.g. 100) for an area or volume conversion instead of squaring/cubing it first. Always write the dimensional factor out explicitly (e.g. $100^2$ ) and calculate it before converting.

03 — Explanation

How do volume and liquid capacity connect?

1 \text{ cm}^3 = 1 \text{ ml} \qquad 1000 \text{ cm}^3 = 1 \text{ litre} \qquad 1 \text{ m}^3 = 1000 \text{ litres}

04 — Explanation

How do I convert time into decimal hours or minutes?

Minutes and seconds use base 60, not base 10 — never treat "1 hour 30 minutes" as $1.3$ hours. Divide the minutes by 60 first: $30 \div 60 = 0.5$ , giving $1.5$ hours.

Exam Tip: Scientific calculators have a DMS (degrees/minutes/seconds) button that converts time formats instantly and avoids this error completely — use it whenever a speed or rate question mixes hours and minutes.

05 — Explanation

How do I work out the formula from a compound unit?

The unit itself tells you the formula — "per" means divide. $\text{kg/m}^3$ means $\text{mass} \div \text{volume}$ , giving density $D = \dfrac{M}{V}$ . The same logic gives speed $S=\dfrac{D}{T}$ and pressure $P=\dfrac{F}{A}$ .

Exam Tip: Before substituting into any compound measure formula, check every value is in the units the formula expects (e.g. density in g/cm³ needs mass in g and volume in cm³, not kg and m³) — convert first if they don't match.

06 — Explanation

How do I convert a speed from km/h to m/s?

Convert distance to metres ( $\times1000$ ) and time to seconds ( $\times3600$ ) together: since $\dfrac{1000}{3600}=\dfrac{1}{3.6}$ , dividing a km/h speed by $3.6$ converts it directly to m/s.

07 — Worked example

Worked Example

A tank measuring $40$ cm $\times 25$ cm $\times 20$ cm is completely filled with water. Find its capacity in litres.

Volume: $40 \times 25 \times 20 = 20{,}000$ cm³.
Convert to litres using $1000$ cm³ $=1$ litre: $20{,}000 \div 1000 = 20$ litres.

A separate car journey covers $90$ km in $1$ hour $12$ minutes. Find the average speed in m/s.

Convert time to hours: $12$ minutes $=12\div60=0.2$ hours, so total time $=1.2$ hours.
Speed in km/h: $90 \div 1.2 = 75$ km/h.
Convert to m/s: $75 \div 3.6 \approx 20.8$ m/s (1 d.p.).